Add $x^2$ and $ \dfrac{3}{5x^2-20} $ to get $ \dfrac{ \color{purple}{ 5x^4-20x^2+3 } }{ 5x^2-20 }$.

Step 1: Write $ x^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 5x^2-20 }$.

$$ \begin{aligned} x^2+ \frac{3}{5x^2-20} & \xlongequal{\text{Step 1}} \frac{x^2}{\color{red}{1}} + \frac{3}{5x^2-20} = \frac{ x^2 \cdot \color{blue}{ \left( 5x^2-20 \right) }}{ 1 \cdot \color{blue}{ \left( 5x^2-20 \right) }} + \frac{ 3 }{ 5x^2-20 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 5x^4-20x^2 } }{ 5x^2-20 } + \frac{ \color{purple}{ 3 } }{ 5x^2-20 }=\frac{ \color{purple}{ 5x^4-20x^2+3 } }{ 5x^2-20 } \end{aligned} $$

Subtract $ \dfrac{8}{x+2} $ from $ \dfrac{5x^4-20x^2+3}{5x^2-20} $ to get $ \dfrac{ \color{purple}{ 5x^4-20x^2-40x+83 } }{ 5x^2-20 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{5x-10}$.

$$ \begin{aligned} \frac{5x^4-20x^2+3}{5x^2-20} - \frac{8}{x+2} & = \frac{ 5x^4-20x^2+3 }{ 5x^2-20 } - \frac{ 8 \cdot \color{blue}{ \left( 5x-10 \right) }}{ \left( x+2 \right) \cdot \color{blue}{ \left( 5x-10 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 5x^4-20x^2+3 } }{ 5x^2-20 } - \frac{ \color{purple}{ 40x-80 } }{ 5x^2-20 }=\frac{ \color{purple}{ 5x^4-20x^2+3 - \left( 40x-80 \right) } }{ 5x^2-20 } = \\[1ex] &=\frac{ \color{purple}{ 5x^4-20x^2-40x+83 } }{ 5x^2-20 } \end{aligned} $$